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## G = M4(2).D10order 320 = 26·5

### 12nd non-split extension by M4(2) of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — M4(2).D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — D4⋊D10 — M4(2).D10
 Lower central C5 — C10 — C2×C20 — M4(2).D10
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for M4(2).D10
G = < a,b,c,d | a8=b2=c10=1, d2=a6, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 446 in 108 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C52C8, C52C8, C40, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4.4D4, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, D4⋊D5, Q8⋊D5, C5×M4(2), C5×D8, C5×SD16, C2×D20, D4×C10, C5×C4○D4, C20.53D4, C20.46D4, C20.D4, C2×D4⋊D5, D4.Dic5, D4⋊D10, C5×C8⋊C22, M4(2).D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.4D4, D4×D5, D42D5, C2×C5⋊D4, Dic5⋊D4, M4(2).D10

Smallest permutation representation of M4(2).D10
On 80 points
Generators in S80
```(1 17 40 21 63 41 59 78)(2 79 60 42 64 22 31 18)(3 19 32 23 65 43 51 80)(4 71 52 44 66 24 33 20)(5 11 34 25 67 45 53 72)(6 73 54 46 68 26 35 12)(7 13 36 27 69 47 55 74)(8 75 56 48 70 28 37 14)(9 15 38 29 61 49 57 76)(10 77 58 50 62 30 39 16)
(1 63)(3 65)(5 67)(7 69)(9 61)(12 46)(14 48)(16 50)(18 42)(20 44)(22 79)(24 71)(26 73)(28 75)(30 77)(32 51)(34 53)(36 55)(38 57)(40 59)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 10 59 39 63 62 40 58)(2 57 31 61 64 38 60 9)(3 8 51 37 65 70 32 56)(4 55 33 69 66 36 52 7)(5 6 53 35 67 68 34 54)(11 12 72 26 45 46 25 73)(13 20 74 24 47 44 27 71)(14 80 28 43 48 23 75 19)(15 18 76 22 49 42 29 79)(16 78 30 41 50 21 77 17)```

`G:=sub<Sym(80)| (1,17,40,21,63,41,59,78)(2,79,60,42,64,22,31,18)(3,19,32,23,65,43,51,80)(4,71,52,44,66,24,33,20)(5,11,34,25,67,45,53,72)(6,73,54,46,68,26,35,12)(7,13,36,27,69,47,55,74)(8,75,56,48,70,28,37,14)(9,15,38,29,61,49,57,76)(10,77,58,50,62,30,39,16), (1,63)(3,65)(5,67)(7,69)(9,61)(12,46)(14,48)(16,50)(18,42)(20,44)(22,79)(24,71)(26,73)(28,75)(30,77)(32,51)(34,53)(36,55)(38,57)(40,59), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10,59,39,63,62,40,58)(2,57,31,61,64,38,60,9)(3,8,51,37,65,70,32,56)(4,55,33,69,66,36,52,7)(5,6,53,35,67,68,34,54)(11,12,72,26,45,46,25,73)(13,20,74,24,47,44,27,71)(14,80,28,43,48,23,75,19)(15,18,76,22,49,42,29,79)(16,78,30,41,50,21,77,17)>;`

`G:=Group( (1,17,40,21,63,41,59,78)(2,79,60,42,64,22,31,18)(3,19,32,23,65,43,51,80)(4,71,52,44,66,24,33,20)(5,11,34,25,67,45,53,72)(6,73,54,46,68,26,35,12)(7,13,36,27,69,47,55,74)(8,75,56,48,70,28,37,14)(9,15,38,29,61,49,57,76)(10,77,58,50,62,30,39,16), (1,63)(3,65)(5,67)(7,69)(9,61)(12,46)(14,48)(16,50)(18,42)(20,44)(22,79)(24,71)(26,73)(28,75)(30,77)(32,51)(34,53)(36,55)(38,57)(40,59), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10,59,39,63,62,40,58)(2,57,31,61,64,38,60,9)(3,8,51,37,65,70,32,56)(4,55,33,69,66,36,52,7)(5,6,53,35,67,68,34,54)(11,12,72,26,45,46,25,73)(13,20,74,24,47,44,27,71)(14,80,28,43,48,23,75,19)(15,18,76,22,49,42,29,79)(16,78,30,41,50,21,77,17) );`

`G=PermutationGroup([[(1,17,40,21,63,41,59,78),(2,79,60,42,64,22,31,18),(3,19,32,23,65,43,51,80),(4,71,52,44,66,24,33,20),(5,11,34,25,67,45,53,72),(6,73,54,46,68,26,35,12),(7,13,36,27,69,47,55,74),(8,75,56,48,70,28,37,14),(9,15,38,29,61,49,57,76),(10,77,58,50,62,30,39,16)], [(1,63),(3,65),(5,67),(7,69),(9,61),(12,46),(14,48),(16,50),(18,42),(20,44),(22,79),(24,71),(26,73),(28,75),(30,77),(32,51),(34,53),(36,55),(38,57),(40,59)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,10,59,39,63,62,40,58),(2,57,31,61,64,38,60,9),(3,8,51,37,65,70,32,56),(4,55,33,69,66,36,52,7),(5,6,53,35,67,68,34,54),(11,12,72,26,45,46,25,73),(13,20,74,24,47,44,27,71),(14,80,28,43,48,23,75,19),(15,18,76,22,49,42,29,79),(16,78,30,41,50,21,77,17)]])`

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 8A 8B 8C 8D 8E 8F 8G 10A 10B 10C 10D 10E ··· 10J 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D order 1 2 2 2 2 2 4 4 4 5 5 8 8 8 8 8 8 8 10 10 10 10 10 ··· 10 20 20 20 20 20 20 40 40 40 40 size 1 1 2 4 8 40 2 2 4 2 2 8 10 10 20 20 20 40 2 2 4 4 8 ··· 8 4 4 4 4 8 8 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 C5⋊D4 D4.4D4 D4×D5 D4⋊2D5 M4(2).D10 kernel M4(2).D10 C20.53D4 C20.46D4 C20.D4 C2×D4⋊D5 D4.Dic5 D4⋊D10 C5×C8⋊C22 C5⋊2C8 C5×D4 C5×Q8 C8⋊C22 C2×C10 M4(2) C2×D4 C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 4 4 2 2 2 2

Matrix representation of M4(2).D10 in GL8(𝔽41)

 21 3 32 11 0 0 0 0 15 6 2 2 0 0 0 0 37 2 38 38 0 0 0 0 27 39 3 17 0 0 0 0 0 0 0 0 2 28 0 15 0 0 0 0 2 28 17 15 0 0 0 0 12 12 0 0 0 0 0 0 37 37 12 11
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3 1 0 1
,
 18 40 17 24 0 0 0 0 36 23 38 0 0 0 0 0 12 16 17 1 0 0 0 0 29 28 40 24 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 34 25 1 9 0 0 0 0 1 0 0 0 0 0 0 0 33 40 39 16
,
 20 24 24 20 0 0 0 0 26 21 3 20 0 0 0 0 0 0 24 38 0 0 0 0 0 0 1 17 0 0 0 0 0 0 0 0 34 25 1 9 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 40 33 39 7

`G:=sub<GL(8,GF(41))| [21,15,37,27,0,0,0,0,3,6,2,39,0,0,0,0,32,2,38,3,0,0,0,0,11,2,38,17,0,0,0,0,0,0,0,0,2,2,12,37,0,0,0,0,28,28,12,37,0,0,0,0,0,17,0,12,0,0,0,0,15,15,0,11],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,3,0,0,0,0,0,40,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[18,36,12,29,0,0,0,0,40,23,16,28,0,0,0,0,17,38,17,40,0,0,0,0,24,0,1,24,0,0,0,0,0,0,0,0,0,34,1,33,0,0,0,0,0,25,0,40,0,0,0,0,1,1,0,39,0,0,0,0,0,9,0,16],[20,26,0,0,0,0,0,0,24,21,0,0,0,0,0,0,24,3,24,1,0,0,0,0,20,20,38,17,0,0,0,0,0,0,0,0,34,0,1,40,0,0,0,0,25,0,0,33,0,0,0,0,1,1,0,39,0,0,0,0,9,0,0,7] >;`

M4(2).D10 in GAP, Magma, Sage, TeX

`M_4(2).D_{10}`
`% in TeX`

`G:=Group("M4(2).D10");`
`// GroupNames label`

`G:=SmallGroup(320,826);`
`// by ID`

`G=gap.SmallGroup(320,826);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,1123,297,136,1684,851,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;`
`// generators/relations`

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